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Updated: Feb 9, 2026

A Microfluidic-based Hydrodynamic Trap for Single Particles

A Microfluidic-based Hydrodynamic Trap for Single Particles

Published on: January 21, 2011

Geometric hydrodynamics via Madelung transform.

Boris Khesin¹, Gerard Misiolek², Klas Modin^3,4

¹Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada; khesin@math.toronto.edu.

Proceedings of the National Academy of Sciences of the United States of America

|May 31, 2018

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We present a geometric framework for studying Newton's equations on infinite-dimensional spaces. This approach naturally describes hydrodynamical partial differential equations and reveals connections between quantum and classical mechanics.

Area of Science:

Geometric mechanics
Mathematical physics
Fluid dynamics

Background:

Newton's equations govern classical mechanics.
Infinite-dimensional spaces are crucial for advanced physics models.
Partial differential equations (PDEs) describe many physical phenomena.

Purpose of the Study:

To introduce a novel geometric framework for Newton's equations.
To analyze these equations on infinite-dimensional configuration spaces.
To connect quantum mechanics (Schrödinger equation) with classical mechanics (Newton's equations).

Main Methods:

Utilizing a geometric framework for infinite-dimensional spaces.
Applying the Madelung transform to link Schrödinger and Newton's equations.

Keywords:

Fisher–Rao Newton’s equations hydrodynamics infinite-dimensional geometry quantum information

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Employing the Fisher-Rao information metric on the space of probability densities.

Main Results:

Newton's equations are naturally described within this geometric framework.
The Madelung transform is identified as a symplectomorphism between phase spaces.
The Madelung transform is also a Kähler map when using the Fisher-Rao metric.

Conclusions:

The geometric framework provides a unified perspective on classical and quantum dynamics.
The Madelung transform offers a powerful tool for analyzing hydrodynamical PDEs.
This work opens avenues for exploring dynamical applications in mathematical physics.